Abstract
The purpose of this paper is to prove that every 2-isometry without any other conditions from a fuzzy 2-normed linear space to another fuzzy 2-normed linear space is affine, and to give a new result of the Mazur-Ulam theorem for 2-isometry in the framework of 2-fuzzy 2-normed linear spaces. MSC: 03E72; 46B20; 51M25; 46B04; 46S40
Highlights
A satisfactory theory of -norm and n-norm on a linear space has been introduced and developed by Gähler in [, ]
Different authors introduced the definitions of fuzzy norms on a linear space
Let S be the space of real numbers, (X) is a linear space over the field S where the addition and multiplication are defined by f + g = (x, μ) + (y, η) = (x + y, μ ∧ η) : (x, μ) ∈ f and (y, η) ∈ g and λf = : (x, μ) ∈ f, where λ ∈ S
Summary
A satisfactory theory of -norm and n-norm on a linear space has been introduced and developed by Gähler in [ , ]. Alaca [ ] gave the concepts of -isometry, collinearity, -Lipschitz mapping in -fuzzy -normed linear spaces He gave a new generalization of the Mazur-Ulam theorem when X is a -fuzzy -normed linear space or (X) is a fuzzy -normed linear space. Park and Alaca [ ] introduced the concept of -fuzzy n-normed linear space or fuzzy n-normed linear space of the set of all fuzzy sets of a non-empty set They defined the concepts of n-isometry, n-collinearity, n-Lipschitz mapping in this space. They generalized the Mazur-Ulam theorem, that is, when X is a -fuzzy n-normed linear space or (X) is a fuzzy n-normed linear space, the MazurUlam theorem holds. A fuzzy subset N of X × R (R, the set of real numbers) is called a fuzzy norm on X if and only if:
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